LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
8.3. HOMOMORPHIC ANALYSIS OF THE SPEECH MODEL 449
r ^ [ n ]
g ^ [ n ]
2
1
0
−1
−2
(a) Glottal Pulse Complex Cepstrum
−5 0
quefrency nT in ms
5
0
−0.2
−0.4
−0.6
−0.8
(c) Radiation Load Complex Cepstrum
−1
−5 0
quefrency nT in ms
5
v ^ [ n ]
p ^ [ n ]
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
(b) Vocal Tract Complex Cepstrum
−0.5
−5 0
quefrency nT in ms
5
1
0.8
0.6
0.4
0.2
0
(d) Voiced Excitation Complex Cepstrum
−0.2
−5 0 5 10 15 20 25
quefrency nT in ms
Figure 8.17: Complex cepstra of the speech model: (a) Glottal pulse ˆg[n], (b)
Vocal tract impulse response ˆv[n], (c) Radiation load impulse response ˆr[n], and
(d) Periodic excitation ˆp[n].
from which it follows that
ˆp[n] =
∞
k=1
β k
k δ[n − kNp]. (8.46)
As seen in Figure 8.17(d), the spacing between impulses in the complex cepstrum
due to the input p[n] is Np = 80 samples, corresponding to a pitch period of
1/F0 = 80/10000 = 8 ms. Note that in Figure 8.17 we have shown the discrete
quefrency index in terms of ms, i.e., the horizontal axis shows nT .
According to Eq. (8.44), the complex cepstrum of the synthetic speech output
is the sum of all of the complex cepstra in Figure 8.17. Thus, ˆs[n] =
ˆhV [n] + ˆp[n] is depicted in Figure 8.18(a). The cepstrum, being the even part
of ˆs[n] is depicted in Figure 8.18(b). Note that in both cases, the impulses due
to the periodic excitation tend to stand out from the contributions due to the
system impulse response. The location of the first impulse peak is at quefrency
Np, which is the period of the excitation. This is the basis for the use of the
cepstrum or complex cepstrum for pitch detection; i.e., the presence of a strong
peak signals voiced speech, and its quefrency is an estimate of the pitch period.
Finally, it is worthwhile to connect the z-transform analysis employed in
this example to the discrete-time Fourier transform representation of the complex
cepstrum. This is depicted in Figures 8.19(a) and 8.19(b) which show